Method and system for hierarchical multi-scale part design with the aid of a digital computer

ABSTRACT

The present disclosure is directed to a method and system for hierarchical multi-scale design with the aid of a digital computer. A hierarchical representation of a shape and material distribution is constructed which satisfies a top-level constraint at a top-level of representation. Properties for families of designs at each of the lower levels of representation that satisfy additional constraints link each of the lower levels of representation to at least a next higher level of the representation.

This invention was made with government support under contract numberHR0011-17-2-0030 awarded by DARPA. The government has certain rights inthe invention.

SUMMARY

The present disclosure is directed to a method and system forhierarchical multi-scale design with the aid of a digital computer. Inone embodiment, a top-level design constraint and a design objective aredefined for an object that is targeted for manufacturing viamanufacturing instrument. The object boundary is divided into a numberof first-level cells. Each cell being assigned one or more propertiesthat collectively satisfy the top-level design constraint. For two ormore levels of a hierarchy, beginning with the first-level cells asparent cells, each of the parent cells are divided into a plurality ofchild cells. The child cells of each parent cells each have childproperties that collectively satisfy the one of more properties therespective parent cells. The child properties within each of the childcells are optimized while still collectively satisfying the one of moreproperties the respective parent cells. The levels are complete when aselected set of child cells represent a design suitable for tolerancesof the manufacturing instrument and that satisfy the top-level designconstraint. The selected set of child cells being used to define amanufacturable design of the object

In another embodiment, at least one design criterion comprising at leastone constraint is specified. A hierarchical representation of familiesof designs is constructed that satisfy the specified criteria at everylevel of the representation. The hierarchical representation includesone or more levels of representation wherein lower levels ofrepresentation provide more details about the designs than the higherlevels of representation, thereby further restrict the design families.Families of designs at every level of the representation aresynthesized, that satisfy the specified criteria as well as additionalconstraints that link the levels of the representation.

In another embodiment, a hierarchical representation of surrogateproperties of shape and material distributions of a family of designs bya spatial cellular decomposition is synthesized with properties assignedto each cell at each level of the hierarchical representation. Startingfrom a top level of the hierarchical representation, properties at thetop level are determined that satisfy constraints specified at the toplevel based on an analysis performed at the top level. For each levelbelow the top level, a process is performed that involves furtherdecomposing each cell at a given level into disjoint cells of the nextlevel. Decisions on the surrogate properties assigned to the child cellsare made based on an analysis performed at the given level subject toadditional constraints that enforce the properties at the cell at thegiven level to remain consistent with the surrogate properties decidedat the cell at the previous level.

These and other features and aspects of various embodiments may beunderstood in view of the following detailed discussion and accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The discussion below makes reference to the following figures, whereinthe same reference number may be used to identify the similar/samecomponent in multiple figures. The drawings are not necessarily toscale.

FIGS. 1 and 2 are schematic diagrams that illustrate examples ofcellular representation schemes for representation of surrogateproperties of designs according to example embodiments;

FIG. 3 is a diagram showing a hierarchical representation of multi-scaledesign classes in a declarative form and the down-scaling processaccording to an example embodiment;

FIGS. 4 and 5 are flowcharts showing a forward and inverse problemsolution according to example embodiments;

FIGS. 6 and 7 are diagrams showing an analysis and computationalworkflow of a single-scale, forward solution according to an exampleembodiment;

FIGS. 8 and 9 are diagrams showing an analysis and computationalworkflow of a single-scale, inverse solution according to an exampleembodiment;

FIGS. 10A and 10B are flowcharts showing a forward and inverse problemsolution according to other example embodiments;

FIGS. 10C and 11 are diagrams showing an analysis and computationalworkflow of a single-scale, forward solution according to anotherexample embodiment;

FIGS. 12 and 13 are diagrams showing an analysis and computationalworkflow of a single-scale, inverse solution according to anotherexample embodiment;

FIGS. 14A, 14B, 15A, 15B, and 15C are diagrams showing an analysis andcomputational workflow of a multi-scale, inverse solution according toanother example embodiment;

FIG. 16 is a diagram showing changes of at-scale properties at differenthierarchies for a design according to an example embodiment; and

FIG. 17 is a block diagram of a system according to an exampleembodiment.

DETAILED DESCRIPTION

The present disclosure relates to designing of mechanical parts formulti-scale criteria. Modern parts are made with increasingly complexstructures and multiple materials at high resolution using techniquessuch as additive manufacturing (AM). The design of such structures fordesired functionality is challenging because of the diverse physicalphenomena occurring at a range of size scales; for instance, from themicrostructure behavior to the emerging bulk behavior at themacro-scale. Analyzing for all of these scales at once is impracticalwith the existing methods because of theoretical and computationallimitations. The state-of-the-art methods tackle the problem byseparating the scales and up-/down-scaling using theories of (forwardand inverse) ‘homogenization’. However, they are all limited by theassumptions on neighborhood size for averaging properties, separabilityof scales, ergodicity, periodicity, linearity, etc.

This disclosure describes systems and methods that provide flexibilityin multi-scale organization and explores a larger design space withadded functional versatility. These systems and methods can generatecomplex designs that are beyond the reach of compute-intensivehigh-resolution single-scale methods as well as multi-scale methodsoperating on restricted material architectures and homogenizationassumptions.

The rapid growth in fabrication technology has enabled enormousstructural complexity in functional parts, using additive, subtractive,or combined processes. High-resolution multi-material 3D printingtechniques, among others, have enabled architecting materials at a levelof detail and complexity that is multiple orders of magnitude finer thanthe part's overall size. The existing computational design tools are,however, unable to explore the immense design space implied by theability to customize parts at such levels of detail and complexityspanning several length scales (e.g., from microns to meters). Theoverall performance of the part, on the other hand, depends on physicalbehavior that occurs at multiple scales, sometimes ranging from thenano-scale to macro-scale phenomena. Analyzing for all of these scalesat once is not only impractical because of computational limitations,but also often meaningless. For example, topology optimization has beensuccessful in designing parts for optimized structural deformation andcompliance at the macro-scale, but the optimized design can fail due tostress concentrations and crack propagation that are not accounted foras they occur at the micro-scale.

High-performance computing (HPC) is rapidly growing more powerful andaffordable. Compute-intensive graphical processing units (GPU) withthousands of processors are arriving at a cost point of a few cents percore, motivating a shift in computational paradigm towards simpler butmore parallelizable data structures and algorithms. However, the lineargrowth of computing power at its peak (illustrated best by Moore's law)is overshadowed by the exponential growth of the design space complexitywith improved manufacturing resolution. Synthesizing designs representedexplicitly at full-resolution is simply intractable, not to mention ahost of other computational problems such as numerical instabilities andconvergence issues that appear in such attempts.

In addition to the computational barriers, it may not be necessary (oreven desirable) to represent and analyze all details at the finestscale. Different physical phenomena and behavioral requirements are onlymeaningful at one observation scale or another, while others are sofundamental that are preserved across the scales. For instance,statistical fluctuations at the micro-scale of a loaded structure maynot be relevant to the mean deformation or strain energy at themacro-scale. Hence, it makes sense to “homogenize” the relevant materialproperties over neighborhoods within the size scale of millimeters andperform finite element analysis (FEA) at the macro-scale, at which suchneighborhoods are small enough discretizations, to design for optimalcompliance.

However, stress concentrations at the tip of micro-cracks are notobserved in the resulting bulk behavior obtained from macro-analysis,though they can have dire consequences that transcend the scales, e.g.,total failure due to crack growth/propagation. In addition tointroducing new failure modes, the small-scale phenomena can beexploited to improve the large-scale performance. Examples of the latterare using metal foams for heat exchanger design, micro-fluidics forlab-on-a-chip design, and micro-fabricated hairs for adhesive design. Inorder to exploit the freedom in choosing complex forms fit into smallspatial neighborhoods as well as to avoid the nontrivial failure modesthat come with such complexities, there is a need for effectivemulti-scale analysis and design tools.

Although a bottom-up approach to design using a detailed analysis at thefinest level of detail and propagating the effects up to the macro-scaleis conceivable, it is computationally intractable and (sometimes)numerically unreliable. A more practical top-down approach to designrequires a hierarchical representation that implicitly describes thepart by its physically-relevant properties at different size scales. Ateach size scale, the part is designed to satisfy (and possibly optimize)the behavioral requirements that are relevant to the physical lawsgoverning that scale. As with every multi-faceted representation, thechallenge is to maintain a consistent representation of the shape andmaterial distribution, so that distinct views at different scales areultimately representing an equivalent physical part or part families tobe manufactured at one scale or another.

Methods and systems described herein represent and design partshierarchically at a finite number of size scales, generally using atop-down design process. At every scale, an inverse problem is solved bychoosing surrogate properties of shape and material distribution, at alevel of granularity defined by the size scale, such that designrequirements as that level are satisfied. This process involves deciding(and possibly optimizing) bulk properties at the coarsest scale firstand postponing decision-making pertaining to higher-order details to thedesign process at finer scales. The same design process is repeatedrecursively, optimizing objective functions and enforcing constraintsboth of which are scale-dependent behavioral requirements.

At every passage from one scale to the next, the process enforcesinvariance of fundamental properties that are scale-agnostic. This isenforced by passing them to the next scale as an additional set ofconstraints so that subsequent decisions do not affect the feasibilityand optimality of prior decisions at the coarser scales. The challengein realizing this method is in the proper choice of invariantproperties, and in re-architecting analysis tools (forward problemsolvers) in such a way that they can reason about macro-behavior from aknowledge of macro-shape and material properties, before arriving at adecision with regards to finer details or their precise realization.

Some preliminary concepts involved in these methods are those of forwardand inverse problems. To systematically synthesize designs to fulfill aset of performance requirements by searching the design space (theinverse problem), one needs the ability to analyze a given design tocompute its behavior (the forward problem), check it against therequired performance criteria, and decide on the next move in the designspace if the requirements are not met. This iterative paradigm isfundamental to most design methods where a closed-form solution for theinverse problem does not exist. For physics-based performancerequirements, the forward problem is typically solved by numericalmethods such as finite element method (FEM), finite difference method(FDM), discrete element method (DEM), cell method, and others. Theinverse problem is solved by iterating over candidate designs andrepeating the analysis, steered by a variety of techniques such asgradient-descent optimization, stochastic/evolutionary optimization,machine learning, and so on. Unlike forward problems, the solution tothe inverse problems is not unique, leaving one with additionalflexibilities with regards to design decisions. Among the most popularsynthesis paradigms are shape and topology optimization (TO). TO seeksto find the shape (geometry and/or topology) and materialdistribution(s) in a given domain that lead to optimized performance fora given cost (e.g., mass) under specified boundary conditions.

Another preliminary concept is that of spatial discretization schemes.Most existing computational tools for physical analysis/synthesis relyon a discretization of the three-dimensional (3D) space intosufficiently small elements (e.g., ‘cells’). The cells can be thought ofas the atomic design features, and their size determines the granularityof the shape and material representation as well as physical simulation.The most popular choice are equally-sized coordinate axis-aligned cubiccells (e.g., voxels) arranged along a uniform 3D grid. However,arbitrary cellular complexes are possible in principle.

Each cell holds one or more design variables—also called decisionvariables-assigned to the entirety of the cell, meaning that there is novariations throughout a single cell. Thus, one needs to choose smallercells (finer granularity) to represent spatial fields in discrete form.The local shape and material properties are deduced from the designvariables; for instance, a binary variable (1 or 0) assigned to a cellcan be used to indicate whether that cell is full or empty,respectively, while a fraction in the range [0, 1] can be used tospecify the cell's volume occupancy ratio (e.g., 0.35 means 35% full).The number of possible fields that can be represented as such is (atleast) exponential in the number of cells. Similarly, one or morevariables-generally scalars, vectors, or tensors—can be assigned tospecify each cell's material properties such as coefficients of physicalconstitutive laws (e.g., elasticity, heat/electrical conductivity, andso on). As such, the shape and material distributions are described by a(discretized) field over the 3D space. For purposes of the followingdiscussion, the term “field” is meant to describe a discretized fieldalso known as a discrete form.

Searching the design space becomes quickly impractical as the set ofdecision variables grows with more choices of base materials (e.g.,material phases) and their gradation/heterogeneity; higher degrees offreedom that capture material anisotropy (e.g., 21 independent variablesneeded to represent the most general elasticity tensor at every cell);and more fine-grained discretization (e.g., more cells) with improved AMresolution; all of which enable more design flexibility to customizemulti-physical behavior (e.g., coupled thermo-elasticity).

One solution to the complexity problem is to conceive a hierarchicalrepresentation of the design at multiple size scales, such that thedesign space exploration is broken into manageable steps. The top levelis a coarse-grained view of the design with relatively large cells and asmall number of representative (e.g., bulk) material properties assignedto them as decision variables. Each cell can be further decomposed intosmaller cells with a more detailed redistribution of material propertiesat the next level to obtain a more fine-grained view of the design. Thescope of decision making is limited at the finer-grained levels, basedon the already-fixed decisions at the coarser-grained levels, so as toreduce the degrees of freedom and fold the computational complexity.

This method uses a fundamental assumption that is often implicit: todecide/optimize the coarse-grained decision variables, one needs theability to (at least partially) predict the resulting coarse-grainedperformance in spite of the incomplete information pertaining tofine-grained details. Thus at each level, it is assumed that an analysistool can predict some “effective” behavior from a knowledge of some“effective” shape and material properties, using a formulation ofphysical behavior that applies to that level's size scale. Thefiner-grained details that will prevail in subsequent design refinementare assumed to be irrelevant to the predictions happening at thecoarser-grained levels, hence can be ignored when making thecoarser-grained design decisions.

Generally, there are two ways to generate complex designs that spanmultiple length scales in representation detail. One of these ishigh-resolution synthesis at-scale. In such methods, the information isdecided at the finest level of granularity and is passed bottom-up toobtain bulk properties and behavior at the coarser-grained levels; forinstance, by averaging/homogenization (“up-scaling”). Given sufficientcomputational resources, one can use the same level of granularity forrepresentation, analysis, and synthesis at the finest-grained scale.Even with today's technology, this approach to automated designoptimization requires expensive high-performance computing (HPC)infrastructure that is only accessible to a few researchers andengineers. Moreover, the suggested optimized designs are far from thefinal form that could be manufactured and would function in practice.This is due to numerical inaccuracy and noise that will inevitably existwhen solving such large-scale systems of equations (e.g., for FEA).

A level-set TO has been developed that relies on assembly-free solutionof the FEA on high-resolution sparse voxelizations (based on OpenVDBdata structure). The assembly-free approach improves the computationalperformance of the FEA, which is the most intensive step, repeated manytimes in the inverse problem solving loop. The approach can handle alarge number of voxels with reliable numerical precision in reasonabletime on affordable computers. However, increasing the resolution tocapture a wider span of length scales would require longer CPU-hours ifthe FEA elements are selected to align with the cells at the sameresolution or granularity. Alternative approaches based on separatingrepresentation (of shape and material layout) and analysis resolutionshave gained popularity over the past few years, because they enablereducing FEA degrees of freedom to manage computational complexity whileretaining (at least some of) the design freedom offered byhigh-resolution representation of the shape and material layout.

Another way to generate complex designs that span multiple length scalesis a top-down approach to multi-scale hierarchical synthesis.Homogenization theory is a tool that provides the necessary conditionsfor the scales to be separable by means of computing “effective”material properties over neighborhoods that sufficiently characterizethe neighborhood's overall (e.g., bulk) behavior via constitutiverelationships at the macro-scale. The premise is that these effectiveproperties can be obtained by independently studying the shape andmaterial distribution at the micro- and meso-scales.

A two-scale design process is thus conceptualized in which the design isfirst optimized at the macro-scale. In density-based TO methods, thedecision variables at the macroscale are ‘homogenized’ materialproperties, e.g., coefficients of constitutive relations that relate theaverage physical quantities as measured by an observer at macro-scale.Next, a detailed structure that respects the desired homogenizedproperty is synthesized at the micro-/meso-scale. This process is oftencalled ‘inverse homogenization’ or (as it is called in a more generalcontext) ‘realization’ of the macro-properties by mico-distributions.Some challenges to these methods are that their validity relies on theassumption of a substantial separation of scales, so that themicro-geometric details do not affect the macro-properties, e.g., thehomogenized properties are sufficient information to perform acoarse-grained analysis directly on the coarse-grained model of theshape and material layout. These methods rely on a rapid mapping frombulk shape characteristics (e.g., volume fraction) to bulk physicalproperties (e.g., Young's modulus) that is often obtained by offlinecomputational experiments under restrictive assumptions. The boundaryconditions, connectivity/continuity, and/or smooth transition from onecell to another can be difficult to guarantee, leading to use furthersimplifying heuristics and assumptions such as periodic cells, closedcells, predefined boundaries, and so on to come up with valid andreasonable designs in a much more restricted design space.

There are two common scenarios in which classical homogenization comeswith reliable guarantees. First when the design is restricted toperiodic cells with repeating boundary conditions, in which case thecells are called representative/repeating volume elements (RVE). Asecond scenario is when the cells are aperiodic (e.g., open-cell foams)but in such a way that their “effective” macro-properties can beobtained as the average—g over a meso-scale neighborhood—of differentbut statistically equivalent microstructures found in that neighborhood,in which case the cells are called statistical volume elements (SVE). Inother words, the fluctuations of shape and material properties as wellas those of the resulting physical behavior are assumed to average out.

The former scenario normally requires regular structures aligned with anorthogonal Cartesian grid along which the RVEs are tiled with no (orgentle) gradation to enable making independent design decisions at themicrostructure level regardless of neighboring cells. If themicrostructure and its boundary conditions change from one cell toanother, the analysis at the macro-scale is only valid up to someapproximation that depends on the size of the RVEs. The latter scenariorequires the existence of a meso-scale that separate the macro- andmicro-scales-often described in terms of their characteristic lengths asL_(micro)<<L_(meso)<<L_(macro). In both cases the substantial separationof scales—e.g., a few orders of magnitude difference in cell size—is anecessity for any rigorous guarantees to hold. This significantlyrestricts the design regimes that this two-scale approach can producewith reliable analysis in the loop.

One approach to synthesizing structures at finer levels of details is togenerate a library of microstructures offline and match the homogenizedproperties (decided at the macro-scale synthesis) with those of themicrostructure; for instance, using a least-squared error minimizationapproach. In these methods, the range of realizable bulk properties bythe elements in the library are imposed as a constraint on theoptimization problem at the macro-scale. Microstructure realization forgiven bulk properties can also be addressed by machine learning. Analternative approach is to simultaneously optimize the tightly coupledmacro- and meso-scale structures by local linearization of objectivefunctions and constraints for the micro-scale around the optimizedvariables at the macro-scales, which is a common recipe to developmulti-scale TO algorithms.

The number of variables that represent the homogenized behavior of asingle cell at macro-scale can be large even for a single-physicsproblem. For example, the elasticity tensor that represents a linearrelationship between Cauchy stress and strain tensors using a smalldeformations model (averaged or interpolated over the cell) has 21independent decision variables. This leads to high-dimensional materialproperty spaces that are difficult to precompute by sampling,regression, or machine learning for inverse homogenization mapping. Tomake the problem manageable, it is common to restrict the materialstructure to orthotropic (12 variables), cubic (3 variable), or mostcommonly isotropic (2 variables: Young's modulus and Poisson's ratio),missing out potential advantages of a fully anisotropic design.

A notable challenge with homogenization-based methods is to guaranteethat neighboring cells connect properly. This is often achieved byrestricting the shapes of microstructures or their boundaries, applyingheuristics to choose them properly from a restricted set of options, orad hoc “gluing” solutions that further restrict the class of attainabledesigns.

The systems and methods described herein enable multi-scale synthesiseven when the scales are not substantially separated in size, andwithout appealing to any additional assumptions on standalonecorrelations between bulk shape and material properties (e.g., volumefraction) and bulk physical response (e.g., average elasticity). Rather,the methods utilize a full analysis that depends on integral propertiesover the cells. The methods provide flexibility to select the number ofdecision variables; namely, “surrogate” properties (e.g., integralproperties) that partially express the shape and material content of acell as well as the cell size at any level. Moreover, the methods do notrely on precomputed libraries of linearized/homogenized behavior thatconceptualize meso-scale material neighborhoods as points (orinfinitesimal neighborhoods) from the macro-scale viewpoint. Rather,they run full analyses on-the-fly by feeding a property-based solver(e.g., a modified form of FEA) on the said surrogate properties. Sincelarger cells require more information content for high-fidelityanalysis, these methods enable trading cell size with number of retainedproperties per cell.

By adopting a “declarative” and property-oriented, rather thanshape-oriented, hierarchical representation throughout the designprocess, these methods do not need to deal with boundary connections atall. The geometric realization of the hierarchically synthesizedproperties will be postponed to as late as possible in the design stage(akin to ‘lazy computation’ in computer science lexicon). Therealization can be postponed even until a manufacturing process isspecified, which restricts the space of realizable shape and materiallayouts depending on the process parameters.

Hereafter, “shape and material distribution/layout” is used in a purelygeometric sense independent of particular physics. It should not beconfused with material properties that are pertinent to specificphysical constitutive laws. For instance, an ideal representation of asingle-scale solid f made of a single homogeneous and isotropic materialcan be given by a binary-valued ‘indicator’ or ‘characteristic’function/field which is a function that assigns every 3D point with avalue of 0 (“empty”) or 1 (“full”), such that the shape of the part isimplicitly defined by the set of all points at which this function isnonzero. An extension of this is to model material gradation atmacro-scale by a real-valued density field; that is, a function thatassigns every 3D point with a greyscale value (including partially emptyor partially full) that can be interpreted statistically; for instance,in terms of its microstructure's porosity or average volume fraction.

The following embodiments utilize a discrete representation of shape andmaterial distribution fields. Mathematically, these can be expressed astopological ‘co-chains’ (or ‘discrete forms’). Intuitively, the 3D spaceis discretized into a finite number of cells to which the binary-valuedindicator or real-valued density (e.g., occupied volume fraction)function values are assigned as-a-whole. To accommodate multiplematerial phases that collectively form the multi-material part, tupleslike (ρ_(Ω1), ρ_(Ω2), . . . , ρ_(Ωn)) can be assigned one or more of then base materials in different proportions to every cell in 3D space.

For example, given three ‘base materials’ A, B, and C with knownproperties in pure medium, a 3-tuple (0.35, 0.43, 0.12) assigned to acell indicates can be interpreted that the cell's volume is occupiedwith 35% material-A, 43% material-B, 12% material-C, and the 10%remainder is empty, without making any assertion about the precisegeometry of the composite microstructure. In a sense, such arepresentation supplies complete information to specify the design fromthe viewpoint of an observer at the macro-scale, where one does not haveto know any additional details about the microstructure to performcoarse physical analysis. However, this representation does not specifysufficient information to lock down the structure at themeso-/micro-scale.

The methods described herein utilize a multi-scale generalization ofinformational completeness. A representation of shape and materialdistribution is said to be “complete” at a given level/scale if it issufficient to infer functional/behavioral aspects that are relevant tothat size scale. This relies on having access to an analysis tool thatcan use this information to reason about behavioral properties even whenthe finer details about the shape and material distribution is leftundecided or uncertain.

In FIGS. 1 and 2, diagrams schematically illustrate single- andmulti-material representation schemes for unmixed (discretely phased)and mixed (spatially graded) materials at one particular scale (e.g.,macro-scale) in 2D used in a method according to an example embodiment.The 2-cells (planar faces) bounded by 1-cells (planar curves)illustrated for the planar shape and material layout in FIGS. 1 and 2generalize to 3-cells (spatial volumes) bounded by 2-cells (spatialsurfaces) in 3D, and are sketched in 2D here for ease of illustration.Note that there is no limit on the number of base materials or the shapeof cells, but most real-world examples can be solved by a few basematerials and uniform polyhedral cells, often further simplified tocoordinate axis-aligned cubic voxels.

To achieve a hierarchical multi-scale representation, assume K>1 to be auser-defined integer that specifies the number of levels (called‘height’) of the hierarchy. For k=1, 2, . . . , K the k^(th) level is adiscrete form representation of shape and material distribution at thatlevel/scale, in a similar fashion as a single-scale representation ofFIGS. 1 and 2. For every cell c_(k) at the k^(th) level, its diameter(the largest distance between any pair of points in the cell) dia(c_(k))is bounded as L_(k)(1−E_(k))<dia(c_(k))<L_(k)(1+E_(k)) where L_(k),E_(k)>0 are constants. These bounds define the size “scale” at which theobserver measures shape and material properties at that level. In asense, these bounds specify the neighborhood size over which thesemeasurements are meaningful for the physical phenomena at that level.

The 1^(st) level is the coarsest, capturing the least detail butoffering the simplest representation with fewest decision variables fordesign. It is also the scale at which design requirements (e.g.,boundary conditions) are specified. The K^(th) level is the finest butmost complex, typically confined to smaller regions of interest. Thepremise is that designing for the entire structure at this level iscomputationally prohibitive, which is why a divide-and-conquer approachis used and the attention is confined to a subset of the (often morecritical) cells that may require further subdivision into finer-grainedsub-cells.

A cell at the k^(th) level (the ‘parent’) can be decomposed into asub-complex of several smaller disjoint cells at the (k+1)^(th) level(the ‘children’). Hence, the cell complex is constantly refined intomore fine-grained elements at subsequent levels. As with thesingle-scale complex, every cell at every level carries a finite numberof design variables for the entire cell. These are most commonlyintegral properties of the material distribution within the cell. Theobserver at the k^(th) level does not see anything more fine-grainedwith respect to the distribution within the cell. These details willonly prevail when looking at the (k+1)^(th) scale and so forth. Everydesign variable at the k^(th) level is interpreted as the parameters ofa design constraint that the variables at the (k+1)^(th) level mustsatisfy. This is a feature of top-down design and will be explained moreshortly.

At every level, the design variables are to be viewed as a “declarative”representation of the design at the next level. In other words, thek^(th) level declares a set of constraints that define a family ofdesigns which satisfy them. At the (k+1)^(th) level, the morefine-grained representation narrows the family down to a smaller subsetof the design space. As illustrated in FIG. 3, this process continuesuntil the family is tightly constrained enough to implicitly represent aclass of interchangeable parts with narrow variations (e.g., interpretedas tolerances), which can be sent for manufacturing.

This method models a “bottom-less” representation of information thatnever arrives at a single-point design, but keeps narrowing down designspaces/families until they are close enough to a point design formanufacturing. Since the constraints declared by a given parent cell aresatisfied by its children, there is a partial ordering of these designfamilies in terms of containment as one keeps refining them. Thehierarchical representation remains sparse, since there is no need toexplicitly specify constraints on grand-children, grand-grand-children,and so forth. They are automatically satisfied due to the transitivityof the constraint relationships. This is schematically illustrated inthe diagram of FIG. 3.

The diagram in FIG. 3 is an abstract view of a hierarchical“declarative” representation of a multi-scale design. At each level, thedesign is constrained by a set of properties that implicitly define afamily of shape and material distribution fields. Passing from one scaleto the next amounts to further constraining the family by adding moreproperties/constraints. This continues until the family is restrictedenough to fall within part interchangeability bounds for manufacturing.

The constraints imposed by the parent cells on their children include(but need not be limited to) subdivision invariance of the parent cell'sproperties when they are distributed among the children. For example, ifa number of density values, collected in a n-tuple φ₁(c_(k)), ρ₂(c_(k)),. . . ρ_(n1)(c_(k))) is assigned to a given cell c_(k) at the k^(th)level to specify volume fractions occupied by each of the available n≥1base materials inside that cell, the tuples assigned to that cell'schildren c_(k+1)* at the (k+1)^(th) level (collectively denoted by theset of cells ch(c_(k))) are constrained by a number equalities thatindicate the total volume assigned to the union of children has to bethe same as the total volume assigned to the parent. In other words, thehierarchical representation is self-consistent across the scales. Thisis described by Equation (1) below.

ρ_(i)(c _(k))=Σ_(c) _(k+1) _(*∈ch(c) _(k) ₎ρ_(i)(c _(k+)1*), for i=1,2,. . . ,n.  (1)

Homogenization theory attempts to reformulate the governing equations ofphysical behavior (e.g., ordinary/partial differential equations) interms of the statistical average over a given neighborhood that is smallenough for this averaging to retain the key aspects of bulk physicalbehavior. Therefore, one can give a sufficiently detailed (for thepurpose of physical analysis) representation of the shape and materiallayout in terms of the integral properties of the shape and materialdistribution restricted to a cell c_(k) at the k^(th) level. Forexample, the volume fraction of each material phase within a given cellin a FEQ mesh is often sufficient to infer physical properties for FEAto produce reasonably accurate results if the mesh is fine enough. Forcoarser cells, accurate computation of integrals for FEA (e.g., toassemble stiffness matrices or enforce Neumann boundary conditions)requires retaining higher-order integral properties of each cell thanjust volume fraction. Similarly, fluid flow or heat transfer problemsare often homogenized in terms of porosity, which is computed similarlyto volume fraction. For more accurate homogenization and upscaling tolarger cells, higher-order properties (e.g., tortuosity) are used.

The precise shape and material distribution “field” is not known at anyparticular scale, but more information about its integral properties(e.g., statistical average) are discovered as one goes down thehierarchical representation. At the lowest level with the finest-graineddecomposition of cells, one gets as close as needed to the continuousfield (the “ground truth”). However, one will never quite arrive at thetrue field values that change from every point to the next, which isarguably an idealization in the limit of infinitely fine-grained cells.

Homogenization provides adequate approximation of the behavior from aknowledge of average shape and material distribution alone when eachcell c_(k) is broken into many smaller child cells. The hierarchy isdescribed by a large branching factor (number of children) andsignificant separation of length scales<<1 hence L_(k+1)/L_(k)<<1 whereL_(k) is the characteristic length at the k^(th) level, defined earlier.The cells at the (k+1)^(th) level need to be small enough from theviewpoint of an observer at the k^(th) level for the average to besufficient information for statistically reliable physical analysis. Ourmethod can be viewed as a generalization of the classical (forward andinverse) homogenization, in which we use higher-order integralproperties than just average properties, including but not limited tomoments, correlation functions, harmonic amplitudes, and so forth.

In order to allow more flexibility in the branching factor of thehierarchical decomposition and moderate separation of length scales, theinformation content of each cell can be described with more than asingle variable such as statistical average of the amount of eachmaterial. To this end, the information content of each cell c_(k) isrepresented with a vector of properties per each base material. Forexample, given n base materials and m properties per material phase, wehave an (n×m) matrix of properties per cell as shown in Equation (2)below.

$\begin{matrix}{{\mu \left( c_{k} \right)}:={\begin{pmatrix}{\mu_{1}\left( c_{k} \right)} \\{\mu_{2}\left( c_{k} \right)} \\\vdots \\{\mu_{n}\left( c_{k} \right)}\end{pmatrix}^{T} = \begin{pmatrix}{\mu_{1}^{1}\left( c_{k} \right)} & {\mu_{2}^{1}\left( c_{k} \right)} & \ldots & {\mu_{n}^{1}\left( c_{k} \right)} \\{\mu_{1}^{2}\left( c_{k} \right)} & {\mu_{2}^{2}\left( c_{k} \right)} & \ldots & {\mu_{n}^{2}\left( c_{k} \right)} \\\vdots & \vdots & \ddots & \vdots \\{\mu_{1}^{m}\left( c_{k} \right)} & {\mu_{2}^{m}\left( c_{k} \right)} & \ldots & {\mu_{n}^{m}\left( c_{k} \right)}\end{pmatrix}}} & (2)\end{matrix}$

For the special case with m=1, i.e., when there is only one degree offreedom per material phase per cell, the most viable integral propertyis the total/average amount of that material (absolute or relative) inthe cell. This property takes various forms and can manifest as volumefraction, average density, average porosity, statistical mean (expectedvalue) in the presence of uncertainty, among others. If m>1, i.e., morethan one property are used per base material per cell as decisionvariables, there are different ways in which the vectors μ_(i)(c_(k))can be chosen. Moreover, if these properties are chosen in such a waythat they exhibit certain useful qualities such as additivity,continuity, completeness, convergence, and behavioral relevance, thesequalities are exploited for multi-scale design.

Additivity is considered first. In order to connect the variables acrossthe scales by parent-child relationships, additivity is retained as inEquation (3) below, which is a generalization of Equation (1) throughcomponent-wise addition of μ_(i)(c_(k+1)*)=(μ_(i) ¹(c_(k+1)*), μ_(i)²(c_(k+1)*), . . . , μ_(i) ^(m)(c_(k+1)*)). As before, this can beinterpreted as an equality constraint imposed by the parent cell on itschildren for top-down design.

$\begin{matrix}{{{\mu_{i}\left( c_{k} \right)} = {\sum\limits_{c_{k + 1}^{*} \in {{ch}{(c_{k})}}}^{\;}\; {\mu_{i}\left( c_{k + 1}^{*} \right)}}},{{{for}\mspace{14mu} i} = 1},2,\ldots \mspace{11mu},{n.}} & (3)\end{matrix}$

As to continuity, the properties change continuously with the actualshape and material distribution field. For example, a small geometric ortopological change in the i^(th) base material layout within c_(k) wouldlead to a small change in μ_(i)(c_(k)) according to some metric in theshape and material domain. As to completeness, the finite sequence ofproperties μ_(i)(c_(k))=(μ_(i) ¹(c_(k)), μ_(i) ²(c_(k)), . . . , μ_(i)^(m)(c_(k))) is assumed to be the first m≥1 terms of a (potentiallyinfinite) sequence of properties that completely describe the i^(th)base material's layout within c_(k). As to convergence, the designvariants within the family of designs that satisfy the first m≥1properties—interpreted as constraints in the next (finer-grained)level—differ only with respect to the unknown truncated terms (μ_(i)^(m+1)(c_(k)), μ_(i) ^(m+2)(c_(k)), . . . ). All of the design variantslie within a bounded design subspace, with respect to some metric (e.g.,Hausdorff distance). The space shrinks to a single-point design as m→∞.Finally, for behavioral relevance, the first m≥1 properties providesufficient information to collectively reason about physical performanceof the design variants within the constrained family, with respect tothe physical phenomena that occur at the k^(th) size scale asillustrated in FIG. 3. In other words, access to a computationalanalysis tool is assumed that can map these geometric properties toreliable physical properties, without a knowledge of more fine-graineddetails at the (k+1)^(th) level.

The last condition is provides the ability to conduct a one-way top-downdesign for behavioral requirements, such that design within theconstrained family in the (k+1)^(th) level does not affect the decisionsalready made in the k^(th) level. This will be elaborated below byrevisiting computational workflows for physical analysis in single- andmulti-scale problems.

In this section, the forward and inverse problems at a single scale arediscussed. To evaluate the physical performance of a given partrepresented at a given scale, one typically solves a boundary valueproblem (BVP) using a numerical method such as finite difference method(FDM), finite element method (FEM), finite volume method (FVM), cellmethod, etc. These methods underlie many of the common numericalanalysis and synthesis tools such as structural analysis (usingFEA/FEM), computational fluid dynamics (CFD), shape and topologyoptimization, and their many variations.

The solution field (e.g., displacement, temperature, etc.) characterizesthe response of the design to initial/boundary conditions (IC/BC), bodyforces, or other stimuli, and is represented by a linear combination ofsimple basis functions in a vector space of functions (e.g., Hilbertspace) that covers all possible solution fields at an appropriate levelof granularity. To find the solution(s) that admit the BCs (among otherpotential constraints) the coefficients of this combination are obtainedby solving a discretized form of the BVP; for instance, by minimizing afunctional determined by the weak form of the BVP when using FEA/FEM.The computational workflow for solving the forward problem 400 can beconceptualized at a high level as a set of steps as shown in theflowchart of FIG. 4 according to an example embodiment.

Inputs 401 a to the procedure include shape and material distributionfields (represented in a discrete form). Inputs 401 b also includeIC/BC, body effects, base material properties (for a finite number ofavailable materials), and other FEA parameters. If there ishomogenization involved, the given shape and material distribution field(geometric information) is mapped 402 to a material property fieldspecific to the physics problem under consideration (e.g., stiffness,conductivity, etc.) using empirical relationships. For example, therelative volume fractions occupied by each of the base materials withineach cell are used to infer an “effective” stiffness or conductivity forthe entire composite cell. A number of surface and volume integrals arecomputed 403 over the domain and its boundary as part of the standardFEA. The integrands are typically linear or algebraic combinations ofmaterial property fields obtained in 402, BCs, and FEA basis functions.

A system of linear equations is assembled 404 from the integralscomputed in 403 whose unknowns represent the physical response of thedesign to IC/BC, body forces, and other stimuli. The physical responsefields and (if needed) their sensitivity fields are computed 405 bysolving the system of linear equations and its differentiation withrespect to design variables. Outputs 406 of this computation 405 includethe response fields (e.g., displacement, temperature, etc.) expressed asa linear combination of FEA basis functions. The outputs 406 may alsoinclude the sensitivity fields (e.g., geometric or topologicalsensitivities) expressed as a linear combination of FEA basis functions.

If the results of the analysis are used to solve an inverseproblem—e.g., to design the shape and material fields in such a way thatthe response fields satisfy certain performance criteria—the performancemeasures (e.g., objective function(s)) are computed 407, augmented by ameasure of the extent to which any design constraints are violated(e.g., penalty function(s)) and gradient-like quantities that show apreferred direction to move in the design space. Outputs 408 of thiscomputation 407 include the objective/penalty function(s) and theirgradient(s).

In FIG. 5, a flowchart shows a computational workflow for solving aninverse problem 500 according to an example embodiment. Inputs 501 ainclude one or more initial shape and material distribution fields(represented as discrete form(s)) to use as the first candidatedesign(s) in the synthesis loop. Inputs 501 b also include IC/BC, bodyeffects, base material properties (for a finite number of availablematerials), and other FEA parameters. The forward problem is solved 502for the candidate design to evaluate the objective function(s),violation of constraint(s), and gradient-like quantities. The quantitiescomputed in 502 are used to decide whether the design criteria is met503 (to exit the loop), and if they are not, how the candidate design isto be changed 504 to improve them. Operations 502 and 503 are repeateduntil the design converges to one or more solutions at which the designcriteria are met. When the design criteria are met, the outputs 505include one or more final shape and material distribution fields(represented as discrete form(s)) that have optimized performance andsatisfy the design constraints.

Although the geometric and physical fields can in principle berepresented in different functional bases. For example, the basis forFEA can be defined over cells of simple and regular shapes organizedalong a regular gird (e.g., voxels) that do not conform to the geometricshape of a particular design represented, for example, by polyhedralsurface or volumetric meshes, cubic B-spline surfaces, or NURBS. A hostof ‘immersed boundary’ methods, including meshless/meshfree FEA havebeen developed to use different bases for geometric representation andphysical analysis, to avoid the difficult mesh generation problem, atthe expense of additional challenges such as enforcing thenon-conforming BCs. On the other hand, iso-parameteric and iso-geometricanalyses methods exist to enable using the same basis for geometricrepresentation and physical analysis, also avoiding meshing andre-meshing.

When the design's shape evolves in a synthesis loop and there is nopreference for a specific basis over another—e.g., there is no fixedgeometry with a conforming mesh to pick as the preferred discretization(e.g., to simplify imposing BCs)—it is common to pick a simple basis(e.g., over a regular grid of voxels) to populate the design space ofshape and material distribution fields as well as the performance spaceof physical fields. The last output depends on the first two outputs(BVP solution and its derivative(s)) and is used to guide a designoptimization loop. It is worthwhile taking a deeper look at step 502. Ittypically takes the form of assigning constitutive relationshipcoefficients (e.g., material properties) to the cells based on theirshape and material layout—e.g., mapping volume fractions of differentmaterials, porosity, tortuosity, etc. (geometric information) tostiffness, conductivity, resistance, permittivity, etc. For the simplestdiscretization scheme with binary-valued voxels (0 for empty and 1 forfull) or integer-valued voxels in case of multiple indexed materials,the shape and material distribution can be mapped to constitutivecoefficients by a simple look-up table. Each available material has aknown elasticity tensor, conductivity tensor, etc. obtained frommultiple experiments and represented in a Cartesian coordinate system.This approach assumes that the cells are small enough to assign auniform distribution over each one of them or use a simple interpolationscheme among the vertices of the cells depending on the choice of thebasis functions.

For graded composites, one needs additional assumptions to provide themapping from shape and material distribution—e.g., the tuple (ρ₁, ρ₂,ρ₃) indicating volume fractions occupied by three materials—to theconstitutive coefficients needed by the analysis. The mapping depends ondetailed microstructure geometry, which is not captured by the volumefraction tuple, as well as how the specific materials mix together andbond at the interfaces. To circumvent fine-grained analysis at thecoarse-grained level, different heuristics are used. For instance, forsome physical phenomena a linear mixing model may suffice as theroughest approximation to how properties mix, e.g., E(ρ₁, ρ₂,ρ₃)≈Eρ₁+E₂ρ₂+E₂ρ₃ to estimate the effective modulus of elasticity of athree-material composite E(ρ₁, ρ₂, ρ₃) from the moduli of elasticity forbase materials denoted by E₁, E₂, and E₃. A linear model may not beappropriate depending on the application or physical domain. Forexample, in the well-known solid isotropic material with penalization(SIMP) method of topology optimization, a cubic polynomial formula iscommonly used to penalize partial volume fractions when crisp boundariesare favored.

More elaborate schemes are used today in microstructure designliterature. For instance, aperiodic stochastic lattices such asadditively manufactured Voronoi foams and k-nearest neighbor (k-NN)foams have been studied to establish a relationship between theirparametric representation (e.g., sampling density and thickness) andeffective unit cell material properties. Similarly, non-parameterizedand topologically optimized microstructures have been studied togenerate precomputed libraries of meta-material properties. To make thehuge space of all possible microstructures manageable, these methodsassume certain symmetries in the structure (at least up to statisticalguarantees) to make the bulk behavior expressible with few variables,e.g., Young's modulus and Poisson's ratio for isotropic mechanicalbehavior. For example, it is experimentally observed that while theYoung's modulus strongly depends on the sampling density, the Poisson'sratio does not vary significantly for open-cell foams under certainconditions.

In FIG. 6, a diagram shows an abstract view of the physical analysis asa mapping from a geometric to a physical representation space. Themapping is from shape and material distribution fields to physicalresponse fields, for a fixed set of specifications such as IC/BC, bodyeffects, base material properties, and a policy to combine base materialproperties into an “effective” cell property for partially occupiedcells and mixed materials, as discussed above. For a fixed set of IC/BC,body effects, base materials, and other specifications, the forwardproblem can be viewed as a mapping between two spaces; namely, thedesign space defined by all possible shape and material distributionfields; and the behavior space defined by all admissible physicalresponse fields. In FIG. 7, a diagram shows a high-level view of thecomputational workflow in FEA analysis, which implements the mapping bycomputing volume and surface integrals, which, in turn, are used toassemble a linear system of equations solved by a standard algorithm.The response is used to evaluate the performance of the design, whether(and how much) it violates any design constraints, and so forth. Thediagram of FIG. 7 is a reasonable representation of how most FEA solversmap a representation of the design (shape and material distributionfields) to a representation of performance (physical response fields).

The inverse problem is to find a shape and material distribution thathas optimized performance and satisfies specified constraints. In theabsence of a closed-form solution, the inverse problem is solved byiteratively solving the forward problem and making changes to the designuntil it satisfies the criteria. In FIG. 8, a diagram shows an abstractview of the design synthesis (inverse problem) as an iterative processin which the physical analysis (forward problem) of FIG. 6 is calledrepeatedly 801, 805 to refine the shape and material distribution 804based on evaluation 802 of the achieved performance and its comparison803 with the desired performance. In FIG. 9, a diagram shows ahigh-level view of the computational workflow in design synthesis byadding a feedback loop to the FEA analysis of FIG. 7. To steer theoptimization into local optima, a gradient-like quantity (e.g.,sensitivity field) 808 is also computed as seen in FIGS. 8 and 9.

The material densities assigned to the cells (e.g., voxels) are thedecision variables that are used to evaluate the objective functions,constraints, and their gradients. These quantities are in turn used tomake changes to the design variables—e.g., add, remove, replace, orchange the density of base materials at the cells depending on how muchthey are contributing to the overall performance measure and howsensitive these measures are to their (hypothetical) variations, ascaptured by FIGS. 8 and 9.

The key takeaway from a closer look at these workflows is that for afixed choice of the basis functions, the information needed to evaluatethe performance determine the class of integral properties that need tobe computed. For example, in the case of FEA, the solution depends onthe coefficient matrix A and stimulus vector B in the assembled systemof equations, which, in turn, depend on the following quantities: thevolumetric integral properties of shape and material distributionfields, Dirichlet BCs, and body effects; and the surface integralproperties pertaining to Neumann BCs. This is not surprising as the FEAis entirely based on integral/weak forms of the BVP. Note that theestimation of effective constitutive properties from distribution fieldvalues are also important only to the extent that affects the integralresults.

In order to separate the scales, it should be understood that theinformation needed for physical analysis are integral properties of theshape and material distribution fields. As long as one has access to therequired minimum information to compute the volume and surface integralsused in assembling the linear system of equations at a granularity thatis adequate at a given size scale, one can (in principle) find thephysical behavior relevant to that size scale. Accordingly, one candirectly design the integral properties at that size scale and postponefinding the fields that have those properties to a later stage—e.g.,design at the finer level or manufacturing process planning.

The next section describes forward and inverse problems at multiple sizescales, organized at multiple levels or granularity. The design space isenormous if the shape and material distribution fields are discretizedat the finest-grained resolution (e.g., enabled by today'shigh-resolution manufacturing). To reduce the number of degrees offreedom, it is common to use a multi-scale synthesis approach based onscale-dependent analysis. The BVP is re-framed in terms of homogenizedproperties (“up-scaling”) by assuming correlations between effectivephysical constitutive law coefficients and statistical average of basematerial distribution in each cell. The synthesis proceeds in theopposite direction, where the analysis is repeated in a loop to designfor homogenized properties. The detailed structure at a finer-grainedscale is obtained by inverse homogenization (“down-scaling”). Thisprocess is rarely extended beyond two scales, a macro-scale and ameso-/micro-scale, depending on the method and its assumptions.

Many existing methods use the notion of “micro” too permissively, evenwhen the cells (sometimes called ‘unit cells’) are too large to justifythe existence of a meso-scale for statistical guarantees. The assumptionthat a single (e.g., tensorial) constitutive relationship (e.g., stressand strain related by a single elasticity tensor) is representative ofthe cell's behavior amounts to viewing every unit cell as a “materialpoint” with a small, effectively infinitesimal, neighborhood that isonly approximated by a single finite element for analysis. This is notalways true, leading to low-fidelity analysis.

A “design for properties” methodology is described herein that does notmake assumptions on the size of the cells at different length scales,does not rely on a single (e.g., tensorial) constitutive equation toapproximate a cell's complicated behavior at a given scale, and does notlimit one to a particular parameterized design regime (e.g., foams)upfront. The decisions on realization of the designed properties orparameterization are postponed as much as possible to a later stage inthe design process, e.g., a manufacturing process planning stage, when aprocess plan is found to fulfill a set of declared properties atdifferent length scales.

The goal is to design for a series of “surrogate” integral propertiesp(c_(k)) in Equation (2) assigned to the cells c_(k) at some k^(th)level of the hierarchical representation that not only satisfy theadditivity, continuity, completeness, and convergence characteristicsdiscussed earlier, but also are physically relevant, meaning thatknowing the values of these properties about the design is sufficient toinfer physical behavior that is relevant at the length scale ofinterest. From a computational point of view, this depends on the typeof the physical analysis tool and what information about the shape andmaterial layout it needs to perform its computations with a reasonabledegree of accuracy and fidelity.

For example, FEA solvers typically attempt to solve weak variationalforms of BVP, which relies on integrating the shape and materialdistribution fields over the supports of basis functions. If theintegrand is represented in terms of a linear combination of the basisfunctions, all such integrals can be expanded as ‘moments’ of the shapeand material distribution fields in the said basis. Moments are theinner products of the shape and material distribution fields with somebasis functions (which are related to the FEA basis function) restrictedover the cells. In this case, moments can serve as a set of integralproperties from which the linear system of equations in FEA can beassembled, without knowing the precise details of the geometry that leadto those moments. In other words, the infinite series of moments carrycomplete information to perform full-fidelity FEA, up to the inevitableapproximation due to FEA discretization. The realization problem, whichis to find the shape and material distribution fields that give rise tothose moments (hence to the resulting performance) can be postponed.Therefore, the computational workflows for solving the forward problem1000 and inverse problem 1020 can be reconfigured at a high level as aset of steps as shown in the flowcharts of FIG. 10A and FIG. 10B,respectively, according to example embodiments.

Note that that FIGS. 4 and 5 are example embodiments of state-of-the-artfor forward and inverse problems. The main difference between theflowcharts shown in FIGS. 4 and 5 and the respective flowcharts of FIGS.10A and 10B, is that the design variables are no longer the distributionfields, but their surrogate (e.g., integral) properties; for example(but not necessarily) volume fraction (zeroth moment) and higher-ordermoments for FEA, correlation functions for statistical physics solvers,harmonic amplitudes for spectral solvers, and so on. On the other hand,the performance variables are no longer the physical response fields,but are bulk properties of such fields observed at a chosen scale.

In practice, one can only represent the information content of a cellwith a finite number of surrogate properties, such as volumetricintegral properties. Fortunately, for proper choice of basis functions,a partial sequence of first m≥1 properties μ_(i)(c_(k))=(μ_(i) ¹(c_(k)),μ_(i) ²(c_(k)), . . . , μ_(i) ^(m)(c_(k))) provides sufficientinformation to approximate the integrals needed for physical analysis(e.g., the lowest-order moments for FEA) that converge to the exactvalues as m→∞ with quantifiable bounds of truncation error.

Surface integration is another fundamental task in FEA which may berequired to impose Neumann BCs in the weak variational form. In asimilar fashion to volumetric distributions, surface distributions(e.g., force, flux, flow, etc.) are also representable as discreteforms. Accordingly, surface integral properties at-scale can be obtained(e.g., as moments of surface properties) without actually knowing theprecise surface distribution at the finest detail.

The diagram in FIG. 12 updates the abstract view of the physicalanalysis presented earlier in FIG. 6 by viewing it as a mapping from thespace of integral properties of the base material distribution fields—asopposed to the fields themselves—to the space of bulk properties ofphysical response fields. Each property set is viewed as a point in thenew design space, which corresponds to an entire family—technically, an‘equivalence class’—of shape and material distribution fields in theoriginal view of the design space in FIG. 6. The process in FIG. 13accordingly updates the high-level view of the computational workflow inFIG. 6, where the burden of volume and surface integration is removedfrom the FEA tool because it no longer has access to a detailedrepresentation of the shape and material layout. The required integralsare computed from a combination of input integral properties (e.g.,linear combination of moments for FEA), which can be approximated bymoment fitting using various quadrature rules.

The diagrams in FIGS. 12 and 13 illustrate the closed-loop synthesisbased on the physics solver in FIGS. 10C and 11. The decision variablesare the integral properties themselves rather than the precise shape andmaterial distribution fields that realize them. As such, the synthesisof shape and material layout is bypassed and postponed as it does notaffect the evaluation of objective function, constraints, and theirgradients within the coarse analysis.

Surrogate properties can be set up as projections of the shape andmaterial distribution fields onto a functional basis, e.g., componentsof the fields in some Hilbert space that is appropriate for the physicsproblem and solver class. The key is to set these properties up in sucha way that they capture increasingly more details of the topology andgeometry of the base material with increasing order of the basisfunction on which they are projected. To leverage this refinement, moreproperties will be kept as the multi-scale design process moves down inscale. For example, the 0^(th) moment—which is the same as volumefraction—can be used to design at the macro-scale while higher-ordermoments are designed at the meso-/micro-scale. In this case, our methodsubsumes existing inverse homogenization methods with the exception thatit uses a surrogate variable (such as the volume fraction at each cell)with the understanding that it is all one needs for a coarse analysis,meaning that volume fraction can be used as a “proxy” for physicalproperties such as stiffness or conductivity if the cells are smallenough to capture their behavior by a single stiffness or conductivitytensor. However, our method extends beyond classical (forward andinverse) homogenization as it enables a trade-off between granularityand number of surrogate properties one retains per cell (e.g.,higher-order moments than volume fraction or high-frequency harmonicsfor spectral analysis) provides flexibility to choose cell size, henceenables a top-down design that starts from coarse-grained cells. Thedecided volume fraction at the coarser-grained scale is passed onto thefiner-grained scales as an constraint, e.g., equality constraintinferred from additivity of the surrogate properties in Equation (3).Deciding the properties that capture more details, e.g., higher-ordermoments or higher-frequency harmonics, are postponed to thefiner-grained levels, e.g., to restrict the detailed microstructureshape. For a given cell c_(k) at the k^(th) level, its children mustsatisfy the equality constraints, one for each base material, as shownin Equation (4) below.

$\begin{matrix}{{{\mu_{i}^{0}\left( c_{k} \right)} = {\sum\limits_{c_{k + 1}^{*} \in {{ch}{(c_{k})}}}^{\;}\; {\mu_{i}^{0}\left( c_{k + 1}^{*} \right)}}},{{{for}\mspace{14mu} i} = 1},2,\ldots \mspace{11mu},{n.}} & (4)\end{matrix}$

Once the most significant surrogate properties of each base material fora parent cell c_(k) is decided (and, potentially optimized) based on thecoarse-grained analysis, the optimization at the finer-grained scale istasked with re-distribution of the fixed “budget” of base materialwithin each parent cell (decided in the coarser-grained synthesis) amongits children c_(k+1)*∈ch(c_(k)). In other words, the total amount of thebase materials per cell at the k^(th) level can change at the k^(th)level synthesis, but remains unchanged at the (k+1)^(th) and subsequentlevels' synthesis. The total material is only exchanged among thechildren within a parent cell but not between them and their cousins,captured by the equality constraint in Equation (4). This link betweenthe scales is generalized as shown in Equation (5), where the index j≥0implies an ordering among the surrogate properties from the mostsignificant to the least significant—e.g., low-order to high-order,low-frequency to high-frequency, and so forth, depending on the context.

$\begin{matrix}{{{\mu_{i}^{j}\left( c_{k} \right)} = {\sum\limits_{c_{k + 1}^{*} \in {{ch}{(c_{k})}}}^{\;}\; {\mu_{i}^{j}\left( c_{k + 1}^{*} \right)}}},{{{for}\mspace{14mu} i} = 1},2,\ldots \mspace{11mu},{n.}} & (5)\end{matrix}$

In general, these methods assume a hierarchical cell decomposition inwhich a partial sequence of at-scale surrogate properties is assigned asunknowns to the cells at different levels. For example, moments obtainedas projections or inner products (in Hilbert space) of the shape andmaterial distribution fields with monomials of various exponents can beordered as shown in Equation (6) below.

$\begin{matrix}{\overset{\overset{1^{st}\mspace{14mu} {level}}{}}{\mu^{0,0,0}},\overset{\overset{2^{nd}\mspace{14mu} {level}}{}}{\mu^{1,0,0},\mu^{0,1,0},\mu^{0,0,1}},{\overset{\overset{3^{rd}\mspace{14mu} {level}}{}}{\mu^{1,1,0},\mu^{1,0,1},\mu^{0,1,1},\ldots \mspace{11mu},\mu^{1,1,1},}\mspace{14mu} \ldots}} & (6)\end{matrix}$

The indexing scheme in this case is based on the exponent of the x, y,and z components of the position in 3D space, respectively. This doesnot assume any particular rules on how the infinite sequence of momentsis divided among the different levels, as long as it makes sense todesign for the level of detail that they provide at the length scaleunder consideration and it can be used in a meaningful fashion by theanalysis at-scale. Any division is acceptable as long as more of themost significant properties are decided in the proper order as thesynthesis proceeds down the scales.

As this happens, the equivalence class of designs associated with theoptimized surrogate properties per cell keeps shrinking as the synthesisproceeds down the scales, as show in FIG. 3. The design at thefinest-grained level of detail is represented by K properties inEquation (1) which are restrictive enough to be realized withinmanufacturing tolerances. The abstract view of the multi-scale synthesisusing at-scale surrogate properties is shown in FIG. 14, which extendsFIG. 12 from single to multi-level top-down design. The computationalworkflow for this process is shown in the diagram of FIG. 15, whichextends FIG. 13 from single to multi-level top-down design.

Every design at the a given level is a discrete form made of a finitesequence of at-scale surrogate properties per cell that “declaratively”represent an equivalence class of shape and material distributionfields. The optimized properties are passed to the next scale asequality constraints (e.g., due to additivity) that need to be satisfiedalongside physical constraints that are specific to the size scale. TheIC/BC are also populated as constraints in terms of the at-scaleproperties of cell boundaries. This process is repeated to narrow downthe family of designs until it is reasonably detailed to be realized(within a narrow band of variation) by an actual shape and materialdistribution, e.g., parameterized microstructures (e.g., foams) ormanufacturable shapes populated by a manufacturing analysis or processplanning method.

The inputs include one or more initial surrogate integral propertiesat-scale of the shape and material distribution at the coarsest-grainedlevel, represented as discrete form(s). This will be used as the firstcandidate design(s) in the synthesis loop at the 1^(st) level in thehierarchical cell complex. Additional inputs include IC/BC, bodyeffects, base material properties (for a finite number of availablematerials), and other solver-specific (e.g., FEA) parameters at the1^(st) level.

At the k^(th) level (for k=1, 2, . . . , K), the forward problem issolved for the candidate design to evaluate the objective function(s),violation of constraint(s), and gradient-like quantities at the k^(th)level using the physics solver that simulates physical behavior at thelength scale range (L_(k)±ϵ_(k)) that characterizes the k^(th) level'svariation of cell sizes. The quantities computed in the forward solutionare used to decide whether the design criteria are met (to exit theloop), and if they are not, how the candidate design represented by theat-scale properties per cell is to be changed to improve them, as shownin the diagram of FIG. 16. In FIG. 16, a partial sequence of at-scaleintegral properties (e.g., moments) are optimized at each level, to apoint that they capture sufficient detail about shape and materiallayout. As one goes down the hierarchy, more properties are discovered,some of which are subject to constraints imposed by parentrepresentation (e.g., additivity). Other surrogate properties and typesof constraints to link the design representation at consecutive levels(parent-child relationships) are conceivable.

The above forward solution and decisions are repeated until the designconverges to a solution at which the design criteria are met. The abovesteps are then repeated for the next level in the hierarchy, e.g., gofrom k to (k+1) and repeat the process for a more limited region ofspacer, to optimize the finer-grained properties of the childrenc_(k+1)*∈ch(c_(k)) of the cells c_(k) over which further refinement isdeemed necessary according to problem-specific criteria. The optimizedproperties are used at the k^(th) scale to populate the BCs for the(k+1)^(th) scale and impose the constraints in Equation (3) or (5) inaddition to the physical constraints relevant to the (k+1)^(th) scale.

The output of this process is one or more final properties at-scale ofshape and material distribution fields at the finest-grained level(represented as discrete form(s)) that have optimal performance andsatisfy the design constraints. This algorithm takes a “lazy evaluation”approach towards determining the actual shape and material layout. At nopoint along the costly optimization loops one requires to know the fullydetailed fields, unlike traditional approaches to synthesis and designoptimization. Once the properties are refined in a top-down processuntil some K^(th) level that is sufficiently detailed, the realizationof the design can be addressed in multiple different ways in apostprocessing step. For instance, if a parameterized microstructureregime is of particular interest—e.g., Voronoi foams, k-NN foams,grain-and-germ textures, and so on—the multi-scale synthesis stops at ascale comparable to the microstructure feature size and calls anpostprocessing algorithm to find a structure that realizes the given setof surrogate properties.

If one or more manufacturing processes are identified, a manufacturingprocess planner can be called to find an as-planned representation ofthe shape and material layout (e.g., a sequence of additive/subtractivemanufacturing actions, G-code, etc.) that realizes the optimized set ofsurrogate properties. The realization of the designed properties as oneor more shape and material layout(s) is postponed until availableparameterized material microstructure or manufacturing process plans arespecified.

A major benefit of the lazy approach is to circumnavigate theconnectivity problem at the RVE boundaries, which is one of the majorchallenges that existing multi-scale methods have to face. When lookingfor a precise realization (e.g., by lattice structures), one has toensure that the nonempty material phases properly connect at theboundaries of the cells to ensure structural integrity. This conditionis often satisfied by restricting the design to materials that areassumed to bond well and disallowing voids altogether at the boundariesof cells. Our method does not have to deal with this problem throughoutthe costly optimization cycles. It rather keeps the options open bydeclarative representation of a maximal family of designs from which asubset of properly connecting microstructures may be found. If no suchsolution exists for a given microstructure parameterization orfabrication method, it will determined during post-processing.

One issue that may be seen using this method is that no tangiblerealization of the at-scale properties are observed (e.g., forvisualization and user inspection) at the intermediate stages, which maybe counterintuitive to the designer. But this is not a true limitation,as one can visualize the family of designs that realize the discoveredsurrogate properties—up to the uncertainty of the undiscovered ortruncated higher-order properties—by the simplest representativerealization in the equivalence class. For instance, if only the 0^(th)moment (e.g., volume fraction) is known for the base materials in agiven cell, it is visualized by using a uniform color along a spectrumof hues that suggest an uncertain distribution. Higher-order moments canbe visualized separately in a similar fashion but will provide lessclues into what to expect of the realized structure.

For a given class of computational physics solvers (e.g., FEA, FVM,statistical physics solvers, etc.) a series of geometric surrogateproperties are determined that provide an “informationally complete”account of shape and material layout for the analysis tool to performits computations unambiguously. These surrogate properties depend on theclass of numerical methods—e.g., solving weak variational form in thecase of FEA requires integration over shape and material properties,hence moments are natural choices as surrogate properties. However,surrogate properties can be physics-agnostic. For instance, the sameproxy variables are common to structural stiffness, heat conduction,compressible fluid flow, and other BVP that can be solved by weakvariational form.

In the example embodiments above, moments were used as a concreteexample of surrogate properties at-scale that can be used directly toperform a wide class of numerical analysis tools (namely, FEA) withoutknowing precise shape and material realization. However, thishierarchical top-down multi-scale design framework is not limited tomoments/FEA per se. There are other examples of properties that can beused with different numerical methods to solve the forward problem. Thismakes them good candidates for decision variables in the inverse problemprovided that gradient-like quantities (e.g., sensitivities) can bereconciled. Other examples include harmonics, such that if each basematerial distribution is viewed as a 3D signal, its discrete Fouriertransform (DFT) is obtained by inner products of the field with harmonicbasis of various frequencies. Fourier expansion is commonly used tosolve BVPs using separation of variables, and is a popularrepresentation for periodic microstructure realization.

Other examples of forward solvers include multi-point correlations andMinkowski functionals. The multi-point correlation functions for one,two, or more points randomly selected within the cell providehigher-order statistical measures of the material distribution for eachbase material within the field. These measures have been used incharacterizing and synthesizing microstructures. The Minkowskifunctionals are morphological descriptors of a domain that captureconnectivity and geometric content of spatial patterns and are useful inquantifying them for statistical physics-based modeling. Minkowksifunctionals are also popular as microstructure descriptors.

To manage the enormous complexity of design for high-resolutionmanufacturing with advanced materials, a single-scale approach tosynthesis is computationally prohibitive. Moreover, different physicalphenomena intrinsically require viewing the same object and measuringits properties at multiple size scales spanning several orders ofmagnitude in units of length.

Existing multi-scale approaches to design are limited by rather strictassumptions that narrow down the possibilities. These range fromimposing a clean separation of scales by limiting the design to smallmicrostructure unit cells (e.g., the “RVE hypothesis”), assuming simpleparametrized structures with reduced constitutive behavior (e.g.,iso-/ortho-tropic foams), assuming periodic building blocks with simpleconnections, and so forth. Many of them come with no guarantees whensome of these assumptions are violated or used too permissively.

A method and system as described above can be configured to performtop-down hierarchical multi-scale design that enables designing abroader range of structures without making such assumptions. The keyidea is to use a declarative approach to directly synthesize theat-scale properties, rather than their precise realization up to thefinest level of detail. At every level of the hierarchy, a family ofequivalent designs are represented by their common properties that aremeasured at that scale, postponing decisions on finer-grained details tothe next level. Once the design family is restricted to a narrow enoughsubset that is unambiguous up to manufacturing tolerances, therealization problem can solved a posteriori (e.g., by manufacturingprocess planning). The key enabling technology is one's ability toanalyze a given family of designs cumulatively using only their commonproperties at-scale. Our approach is to look inside different classes ofnumerical analysis tools and find out the sequence of integralproperties that provide an informationally complete input for them tooperate (e.g., moments for FEA). These properties are used as decisionvariables in design optimization.

An example system utilizing these methods is shown in FIG. 17. Thesystem 1700 includes one or more processors 1701, memory 1702 (e.g.,random access memory) and persistent storage 1704 (e.g., disk drive,flash memory). These components 1701, 1702, 1704 are coupled by datatransfer lines (e.g., memory busses, input/output busses) and areconfigured to perform functions indicated by one or more design modules1706. The processor 1701 may include any combination of centralprocessing units and graphical processing units. In this case, thesystem 1700 includes one or more functional modules 1705 havinginstructions that run on the processor 1701 and may be loaded from andstored to memory 1702, 1704.

The module 1705 receives input data 1706 that defines, e.g., a designdomain/region in the space, design criteria such as specification ofobjective function(s) and constraint(s) at the top-level (e.g., thecoarsest-grained scale), one or more initial designs, available basematerials, IC/BC, and so on. The module 1704 carries a hierarchicalrepresentation of the designs discretized at each scale into a number ofdisjoint cells for the design representation, starting at thecoarsest-grained detail. Each cell is assigned one or more surrogateproperties of shape and material distribution that declarativelyrepresent design families that collectively satisfy the top-levelconstraints.

For one or more levels in the hierarchical design representation,designs are refined iteratively in a top-down order. Beginning with thefirst-level cells at the coarsest-grained scale as parent cells, themodule 1704 divides each of the parent cells into a plurality of childcells, the child cells of each parent cells each having child propertiesthat collectively satisfy the one or more constraints imposed by theproperties of the respective parent cells. The child properties withineach of the child cells are optimized while still collectivelysatisfying the one or more properties of the respective parent cells.This may produce, for example, a plurality of consistent representationsof designs at multiple scales 1710, which define increasingly morerestricted families of feasible designs. Design families that satisfythe design criteria and have the best performance (e.g., determined viaa forward solver used iteratively within the module 1704) can be used asthe basis for the next iteration. The process is repeated in the nextlevel of the hierarchical design representation (at a finer-grainedscale) within one or more of the parent cells, by applying designcriteria imposed partially by the input 1706 and partially by theconstraints obtained from the designed properties at the previous level(parent cells) 1704.

The multi-scale design process is complete when a selected set of childcells represent designs that satisfy the top-level design criteria andis fine-grained enough for realization 1714. The surrogate propertiesobtained at the proper scale for a selected set of child cells are usedto realize one or more shape and material layouts that satisfy them. Therealization 1714 can take the form of computing parameters formicrostructure synthesis in a specific regime (e.g, foams) ordetermining a manufacturing process plan to produce part(s) that satisfysurrogate properties. The design(s) 1716 may be in a format that issuitable to be used as an input to a microstructure synthesis module ormanufacturing process planner, and thereby realize or produce the object1714, e.g, via an additive manufacturing instrument.

In summary, a system is described for designing parts that have desired(e.g., optimized) surrogate properties while satisfying requiredconstraints at a plurality of size scales with the aid of a digitalcomputer. The computer may include a storage device storing parametersspecifying a finite number (K≥1) of levels at different size scales(denoted by the k^(th) scale for k=1, 2, . . . , K) at which the designwill be represented and computed on for different requirements. In thisexample, k: =1 corresponds to the coarsest scale (e.g., macro-scale)typically the one at which design requirements are specified, and k:=Kcorresponds to the finest (e.g., micro- or nano-scale) typically the oneat which design is represented in full-detail. There may be 0, 1, ormore scales in between (e.g., meso-scale). The computer may storeparameters specifying limits (e.g., min/max) on the characteristiclength of cells (defined below) at each size scale.

The computer may store a hierarchical representation of shape andmaterial distribution as a spatial cellular decomposition of theunderlying space in which the part is immersed. This could includemulti-level voxelization—every voxel/cell at one level expands toanother grid of smaller voxels at lower scale. The cell complexes may beof arbitrary shape (e.g., hex mesh, etc.). Each cell is assigned to aspecific size scale and is further decomposed into smaller disjoint,mutually exclusive, cells at the finer size scale. The hierarchicalrepresentation may include geometry (size and shape information ofcells) and topology (adjacency and incidence relations of cells witheach other) at each size scale. The hierarchical representation mayinclude parent/child relationships in the hierarchy that specify whichcells at the (k+1)^(th) (finer-grained scale) collectively unify into acell at the k^(th) scale (coarser-grained scale). The hierarchicalparameters may include values of a finite list of design variables(called “surrogate” properties) assigned to the entirety of each cell torepresent the measurable properties of a designed part's structure asseen by an observer at a given scale. The hierarchical representationmay further include values of a finite list of dependent at-scalephysical properties assigned to the entirety of each cell.

The computer is further configured to store specifications of theintended geometric and physical relationships across the scales, tospecify invariance relations that must hold between the design variablesassigned to parent/child cells of consecutive levels in the hierarchy(e.g., additivity of integral properties); and constraints on shape andmaterial distribution to be enforced on all cells in individual scalesor across cells in a range of scales, including equality and inequalityconstraints on functions of the surrogate properties (e.g., continuityor differentiability of design variables).

The computer is further configured to store parameters defining thesemantics for interchangeability of the part's representation atdifferent scales to qualify invariance in the presence of errors/noise,including: tolerance specifications on cell geometry (e.g., GD&T or itsequivalent); and allowable variations/error margins for design variablesand for resulting properties (e.g., material and physical fields).

The computer further has a processor and memory within which code forexecution by the processor is stored to perform a cell size validitytest. This test involves checking if the size of the cells at each scaleis within the specified limits and flag the ones that require to beresized. The processor performs a consistency test across the scales.This involves iterating over all cells over all scales and testing ifthe invariance relations between parent/child cells (with given designvariables) hold across the scales with respect to interchangeabilitycriteria—e.g., for additive properties, check if the sum of variables atall child cells is equal (up to specified tolerance for errors/noise)with the corresponding variable at their parent cell.

The processor can further perform constraint validation across thescales. This involves iterating over different cells over various scalesand testing if the specified constraints over one scale or acrossmultiple scales (with given design variables) hold with respect tointerchangeability criteria. For example, for continuity, the cells arechecked to determine if they are smaller than a length-threshold carryintegral properties smaller than a value-threshold as prescribed byallowable variations.

The processor can further perform bottom-up evaluation (forwardproblem). Starting from the known design variables at the K^(th) scale(the finest-grained scale), the design variables at the k^(th) scale(coarser-grained scale) are iteratively evaluated by applying specifiedinvariance laws to the design variables at the (k+1)^(th) scale (nextfiner scale) until it reaches the 0-th scale (i.e., the coarsest-grainedscale)—e.g., for additive properties, compute the summation of thevariables at all child cells to obtain the corresponding variable attheir parent cell.

In some embodiments, the storage device further stores a specificationof performance requirements and objectives at a plurality of sizescales, including parameters to specify design/fitness criteria,cost/objective functions, available resources, and other structural,behavioral, or functional requirements to be optimized (e.g., minimized,maximized, or traded off in multi-objective scenarios). The device canalso store constraints on physical properties to be enforced on allcells in individual scales or across cells in a range of scales,including equality and inequality constraints on functions of saidproperties (e.g., continuity of a property with respect to a variable).These constraints include the weak form of initial/boundary valueproblem specification for single- or multi-physics simulation including:IC/BC or their combinations distributed over boundary cells at a givenscale; physical governing equations in the form of discretizedordinary/partial differential equations (turned into algebraic forms);composed of single- or multi-physical conservation (e.g., balance andequilibrium) laws, phenomenological (e.g., constitutive) laws, energyconversion relations, and defining equations; and empirical relationsthat map the design variables into intermediate properties that areneeded to compute the physical properties needed to solve the above.

In such a case, the processor can further perform forward problemsolving (e.g., physical analysis). For example, an FEA solver can becalled at any the relevant size scale(s) to compute the physicalresponse and evaluate performance criteria such as objectivefunction(s), violation of (or lack thereof) constraint(s) (e.g., penaltyfunction(s)), and gradient-like quantities (e.g., sensitivities) of theobjective/penalty function(s) to changes in decision variables. Thesolver can evaluate whether the specified set of objective functions arelocally optimal (or Pareto-optimal for multi-objective cases whichinvolves enumerating solutions along Pareto front) and whether thespecified set of behavioral constraints are satisfied at a given sizescale. The solver can evaluate the proposed change(s) to the designvariables at a given size scale to move closer to an optimized statewith satisfied constraints.

In some embodiments, the data storage further stores a specification ofmanufacturing requirements and objectives at a plurality of size scales,including parameters to specify design/fitness criteria, cost/objectivefunctions, available resources, and other structural, behavioral, orfunctional requirements to be optimized (e.g., minimize, maximized, ortraded off in multi-objective scenarios). The device may further storeconstraints on process properties to be enforced on all cells inindividual scales or across cells in a range of scales, includingequality and inequality constraints on functions of said properties(e.g., continuity of a property with respect to a variable). Theseconstraints include, but are not limited to various fabricationprocess/capability specifications for additive, subtractive, and/orhybrid (e.g., combined additive and subtractive) manufacturingsimulation including: conditions on geometric and material distributionsuch as the minimum manufacturable neighborhood (e.g., tool insert insubtractive processes and molten filament shape in 3D printing),avoiding large overhang angles for support-free 3D printing, properdraw/parting angles for casting/molding, existence of passages to emptythe powder in power-based additive processes, and other process-specificconstraints; conditions on the machine degrees of freedom (e.g.,combinations of translations and rotations); and constraints drawn fromexperimental observation of process-specific material structure such asmicrostructure shape/composition in metal additive manufacturing,relationship between temperature history and structural properties, andso on.

The processor further may further perform forward problem solving(analysis). This may involve calling a manufacturability analysis solverat the relevant size scale(s) to evaluate performance criteria such asobjective function(s), violation of (or lack thereof) constraint(s)(e.g., penalty function(s)), and gradient-like quantities (e.g.,sensitivities) of the objective/penalty function(s) to changes indecision variables. The solver may determine whether the specified setof objective functions are locally optimal (or Pareto-optimal formulti-objective cases) and whether the specified set ofmanufacturability constraints are satisfied at a given size scale. Thesolver may determine whether the proposed change(s) to the designvariables at a given size scale to move closer to an optimized statewith satisfied constraints.

In some embodiments, the storage device further includes specificationof optimization parameters including but not limited to terminationcriteria, penalty factors, etc. In such a case, the processor mayfurther perform inverse problem solving (synthesis). This involvescalling an optimizer (e.g., gradient-based or stochastic) at any therelevant size scale(s) to iteratively call the performance analysissolver. The inverse problem solving further involves applying theproposed change(s) to the design variables at a given size scale to movecloser to an optimized state with satisfied constraints.

The processor may be further configured to pass the optimal results atthe k^(th) scale to initialize the decision variables and specify theconstraints they impose on the decision variables at the (k+1)^(th)scale, and repeat the optimization at the finer-grained scale. Theprocessor may then call a post-processor to realize the optimizedproperties by determining at least one the following: parameterizedmicrostructures at a given scale using either pre-determinedrelationship between parameters and desired properties, or usingoptimization to match them (e.g., in a weighted least-squares sense);and manufacturing process plans such as a sequence ofadditive/subtractive actions, G-code, etc. whose as-manufactured outcomesatisfies the desired properties.

In some embodiments, the storage device further stores a specificationof optimization parameters including but not limited to terminationcriteria, penalty factors, etc. In such a case, the processor mayfurther perform inverse problem solving (synthesis). The inverse problemsolving involves calling an optimizer (e.g., gradient-based orstochastic) at any the relevant size scale(s) to iteratively call themanufacturability analysis solver described above; and apply theproposed change(s) to the design variables at a given size scale to movecloser to an optimized state with satisfied constraints. The processorpasses the optimal results at the k^(th) scale to initialize thedecision variables and specify the constraints they imposed on thedecision variables at the (k+1)^(th) scale, and repeat the optimizationat the finer scale. The processor then calls a post-processor to realizethe optimized properties by determining at least of the following:parameterized microstructures at a given scale using eitherpre-determined relationship between parameters and desired properties,or using optimization to match them (e.g., in a weighted least-squaressense); and manufacturing process plans such as a sequence ofadditive/subtractive actions, G-code, etc. whose as-manufactured outcomesatisfies the desired properties.

The various embodiments described above may be implemented usingcircuitry, firmware, and/or software modules that interact to provideparticular results. One of skill in the arts can readily implement suchdescribed functionality, either at a modular level or as a whole, usingknowledge generally known in the art. For example, the flowcharts andcontrol diagrams illustrated herein may be used to createcomputer-readable instructions/code for execution by a processor. Suchinstructions may be stored on a non-transitory computer-readable mediumand transferred to the processor for execution as is known in the art.The structures and procedures shown above are only a representativeexample of embodiments that can be used to provide the functionsdescribed hereinabove.

Unless otherwise indicated, all numbers expressing feature sizes,amounts, and physical properties used in the specification and claimsare to be understood as being modified in all instances by the term“about.” Accordingly, unless indicated to the contrary, the numericalparameters set forth in the foregoing specification and attached claimsare approximations that can vary depending upon the desired propertiessought to be obtained by those skilled in the art utilizing theteachings disclosed herein. The use of numerical ranges by endpointsincludes all numbers within that range (e.g. 1 to 5 includes 1, 1.5, 2,2.75, 3, 3.80, 4, and 5) and any range within that range.

The foregoing description of the example embodiments has been presentedfor the purposes of illustration and description. It is not intended tobe exhaustive or to limit the embodiments to the precise form disclosed.Many modifications and variations are possible in light of the aboveteaching. Any or all features of the disclosed embodiments can beapplied individually or in any combination are not meant to be limiting,but purely illustrative. It is intended that the scope of the inventionbe limited not with this detailed description, but rather determined bythe claims appended hereto.

1. A computer-implemented method comprising: specifying at least one design criterion comprising at least one constraint; constructing a hierarchical representation of families of designs that satisfy the specified criteria at every level of the representation, the hierarchical representation comprising one or more levels of representation wherein lower levels of representation provide more details about the designs than the higher levels of representation, thereby further restrict the design families; and synthesizing families of designs at every level of the representation that satisfy the specified criteria as well as additional constraints that link the levels of the representation.
 2. The method of claim 1, wherein the design criteria further comprise at least one objective function that provides a figure of merit for designs that satisfy the constraints, the synthesizing the families of designs further comprising optimization with respect to the objective function.
 3. The method of claim 1, wherein the hierarchical representation comprises of surrogate properties of shape and material distributions for the families of designs, the surrogate properties providing sufficient information to evaluate the families of designs against the design criteria.
 4. The method of claim 3, wherein the surrogate properties comprise finite subsequences of a convergent sequence or a series of variables that represent the projection of the shape and material distributions at a given level onto a functional basis.
 5. The method of claim 4, wherein a longer subsequence of variables is decided at lower levels of the hierarchical representation.
 6. The method of claim 3, wherein each level of the hierarchical representation comprises a spatial cellular decomposition wherein every cell at every level of the spatial cellular decomposition is assigned with one or more surrogate properties.
 7. The method of claim 6, wherein the cells at each of the levels are decomposed into disjoint cells to obtain the cells at a next lower level.
 8. The method of claim 6, wherein each level of the hierarchical representation describes the shape and material distributions at a size scale determined by the size of the cells.
 9. The method of claim 6, wherein additional constraints link the properties of a cell at a given level to properties of the cells obtained from the decomposition.
 10. The method of claim 9, wherein the surrogate properties are integral properties of the shape and material distributions within a cell and the additional constraints that link them across the levels are of additivity of integral properties.
 11. The method of claim 3, wherein the final shape and material distributions are not determined until the surrogate properties are synthesized for all levels of representation.
 12. A system comprising: a processor coupled to memory, the memory storing instructions executable by the processor for performing: synthesizing a hierarchical representation of surrogate properties of shape and material distributions of a family of designs by a spatial cellular decomposition with properties assigned to each cell at each level of the hierarchical representation; starting from a top level of the hierarchical representation, determining properties at the top level that satisfy constraints specified at the top level based on an analysis performed at the top level; for each level below the top level, performing a process comprising further decomposing each cell at a given level into disjoint cells of the next level, decisions on the surrogate properties assigned to the child cells being made based on an analysis performed at the given level subject to additional constraints that enforce the properties at the cell at the given level to remain consistent with the surrogate properties decided at the cell at the previous level.
 13. The system of claim 12, wherein the surrogate properties synthesized at each level declaratively represent a family or equivalence class of designs that possess those surrogate properties, and wherein the decisions at each level further restrict the class of designs into subclasses by redistributing the properties of a cell in the previous level.
 14. The system of claim 12, wherein the process terminates when the family or the equivalence class of designs is restricted enough at the bottom level of the representation to be interpreted as a class of interchangeable designs realized by at least one of given manufacturing processes and material microstructures.
 15. The system of claim 14, wherein the given manufacturing process or material microstructure is finalized after the synthesis of surrogate properties is complete at all levels.
 16. The system of claim 12, wherein the processor further optimizes of the surrogate properties at one or more of the levels according to one or more objective functions.
 17. The system of claim 16, wherein the optimization comprises enumerating solutions along Pareto fronts.
 18. The system of claim 12, wherein the decisions at each level are performed by iterations over one or more candidate families of designs represented by surrogate properties at the given level.
 19. The system of claim 18, wherein the iterations are guided by sensitivities or gradients of at least one of the constraints or objective functions at the given level to changes in the surrogate properties.
 20. A method comprising visualizing a hierarchical representation of surrogate properties of shape and material distributions of a family of designs by a spatial cellular decomposition with properties assigned to each cell at each level of the hierarchical representation; and displaying a color-coding of the values of surrogate properties at specified levels of the representation. 